In the Costas frequency coding scheme, the columns represent *M* contiguous time slices (each of duration *t*
_{
b
}) and the rows represent *M* distinct frequencies, equally spaced by *Δ*
*f*
_{0}. We use this scheme for the FDA, namely, only one carrier frequency is transmitted by any one of he *M* FDA elements and each carrier frequency is used only once. The construction algorithms for Costas signals were discussed by Golomb and Taylor [29]. The coding sequence, the order of used frequencies is a concise way to describe the coding matrix. With regard to the difference matrix, note that the top row and the leftmost column are headings and not part of the matrix. The element of the difference matrix in row *i* and column *j* is

$$ D_{i, j}=a_{i+j}-a_{j}, \; i+j\le M, $$

(13)

where *a*
_{
i
} is the *i*th element of the coding sequence. The remaining locations (where *i*+*j*>*M*) are left blank. This equation implies that the first row is formed by taking differences between adjacent elements in the coding sequence, the second row is formed by taking differences between next-adjacent elements, and so on.

Construction of Costas codes can be understood as a construction of stepped-frequency waveforms, where the pulse width *τ* is divided into *M* sub-pulses, each of width *τ*
_{1}. Within each group of *M* sub-pulses, the frequency is increased by *Δ*
*f* from one subpulse to the next. The total signal bandwidth is (*M*−1)*Δ*
*f* and there are *M* sub-pulses; each subpulse has the duration of 1/*Δ*
*f*, then the time-bandwidth product of the transmitted signal is (*M*−1)*Δ*
*f*×*M*×1/*Δ*
*f*=(*M*−1)*M*. Costas codes are similar to stepped-frequency waveforms, except that the frequencies for the subpulse are selected in a random fashion, according to some predetermined rule or logic. Figure 4 compares the hopping orders of LFM and Costas coding schemes, where the *x*-axis and *y*-axis denote the time and frequency, respectively.

The normalized complex envelope of the Costas signal can be expressed as [30]

$$ s(t)=\frac{1}{\sqrt{M\tau_{1}}}\sum\limits_{m=0}^{M-1} u_{m}(t-m\tau_{1}), $$

(14)

where *u*
_{
m
}(*t*)= exp(*j*2*π*
*f*
_{
m
}
*t*), 0≤*t*≤*τ*
_{1}. Note that the sub-pulses are separated in time-domain. It is easily understood that the hopping order strongly affects the ambiguity function of the signal. The ambiguity function can be predicted roughly by overlapping a copy of the binary matrix on itself and then shifting one relative to the other according to the desired delay and Doppler. The corresponding ambiguity function of the matched filter is

$$ {\begin{aligned} \chi(\tau,\nu)=&\frac{1}{M}\sum\limits_{m=0}^{M-1}\exp(j 2\pi m \nu \tau)\\ &\times \left\{\Phi_{mm}(\tau, \nu)+\sum\limits_{q=0,q\neq m}^{M-1}\Phi_{mq}(\tau-(m-q)\tau_{1}, \nu)\right\}, \end{aligned}} $$

(15)

where

$$ \Phi_{mq}(\tau, \nu)=\left(\tau_{1}-\frac{|\tau|}{\tau_{1}}\right)\frac{\sin\alpha}{\alpha}\exp(-j\beta-j2\pi f_{q}\tau) $$

(16)

$$ \alpha=\pi(f_{m} - f_{q} -\nu)(\tau_{1}-|\tau|) $$

(17)

$$ \beta=\pi(f_{m} - f_{q} -\nu)(\tau_{1}+|\tau|). $$

(18)

As noted in Eq. (14), in the standard Costas sequence the sub-pulses corresponding to each element are not aligned. However, in the FDA the transmitted signals from all the elements should be aligned in time-domain; otherwise, the beampattern of the whole array will be decided mainly by some particular elements for a given instant. To avoid this problem, we use the Costas-sequence modulated frequency offsets in a time-aligned way. Taking the Costas sequence illustrated in Fig. 4 as an example, the adopted frequency indexes in the seven time intervals are {4,7,1,6,5,2,3}. Accordingly, our method allows for a seven-element FDA and the seven elements use the frequency offsets 4*Δ*
*f*
_{0}, 7*Δ*
*f*
_{0}, 1*Δ*
*f*
_{0}, 6*Δ*
*f*
_{0}, 5*Δ*
*f*
_{0}, 2*Δ*
*f*
_{0}, and 3*Δ*
*f*
_{0} with *Δ*
*f*
_{0} being the hopping frequency step, respectively. That is, similar to conventional FDA, all the signals are transmitted simultaneously from the Costas modulated FDA.

The frequency fed to the *m*th element of the FDA using Costas-sequence modulated frequency offsets can be generally written as

$$ f_{m}=f_{0}+\Delta f_{m}, \; m=0, 1, 2, \ldots, M-1, $$

(19)

where *Δ*
*f*
_{
m
} is the frequency increment for the *m*th element. The transmitted signal of the *m*th element can then be expressed as

$$ s_{m}(t)=\exp\left(j2\pi \left(f_{0}+\Delta f_{m}\right)t\right). $$

(20)

For an ideal point target at the range *r* and azimuth *θ*, the received echo corresponding to the *m*th antenna is

$$ r_{m}(t)=s_{m}\left(t-2\frac{r+m d\sin\theta}{c_{0}}\right). $$

(21)

By demodulating the received returns with the transmit signal, we can get the baseband signal:

$$ b_{m}(r, \theta)=\exp\left\{-j\frac{4\pi}{c_{0}}\left(f_{0}+\Delta f_{m}\right)\left(r+m d\sin\theta\right)\right\}. $$

(22)

For notation convenience, the above equation can be simply rewritten as

$$ b_{m}(r, \theta)=\exp\left\{-j\phi_{m}\right\}, $$

(23)

where *ϕ*
_{
m
} is

$$ \phi_{m}=\frac{4\pi}{c_{0}}\left(f_{0}+\Delta f_{m}\right)\left(r+m d\sin\theta\right).$$

(24)

In the single snapshot case, the received noise-free echo of one ideal target can be represented as the following receive steering vector

$$ \mathbf{b}(r, \theta)=\left[\begin{array}{cccccc} e^{-j\phi_{0}} & e^{-j\phi_{1}} & \ldots & e^{-j\phi_{m}} & \ldots & e^{-j\phi_{M-1}} \end{array}\right]^{T}, $$

(25)

where ^{T} is the transpose operator. For the multi-target case, the received echo vector for the *k*th snapshot can then be expressed as

$$ \mathbf{x}(k)=\sum\limits_{p=1}^{P} \alpha_{p}(k)\mathbf{b}(r_{p}, \theta_{p}) +\mathbf{n}(k), k=1, 2, \ldots, K, $$

(26)

where *α*
_{
p
}(*k*), *r*
_{
p
}, and *θ*
_{
p
} are the reflection coefficient, slant range, and azimuth angle for the *p*th target at the *k*th snapshot, respectively, *P* is the target number, *K* is the snapshot number, and **n**(*k*) is the *M*×1 additive receiver noise vector. Note that the target reflection coefficient *α*
_{
p
}(*k*) may vary from shapshot to snapshot [31].

Adaptive beamforming algorithms can be used to optimally design the weighting vector **w** to synthesize the desired transmit-receive beampattern. Specifically, when the non-adaptive beamforming algorithm is adopted, the weighting vector is

$$ \mathbf{w}=\mathbf{b}(r_{0}, \theta_{0}), $$

(27)

where *θ*
_{0} and *r*
_{0} denote the angle and range of the desired target, respectively. In this case, the maximum is steered to the expected location (*r*
_{0},*θ*
_{0}). The FDA radar transmit-receive beampattern can then be expressed as

$$ G(r, \theta)\triangleq\ \frac{|\mathbf{b}^{H}(r_{0}, \theta_{0}) \mathbf{b}(r, \theta)|^{2}}{||\mathbf{b}(\theta_{0}, r_{0})||^{4}}. $$

(28)

It is noticed that, like a phased-array radar, the FDA radar has coherent transmit processing gain; however, the FDA radar directional gain depends on both the range and angle parameters, whereas the phased-array radar directional gain depends only on the range parameter. This range-angle-dependent beam provides a potential approach to suppress range-dependent interferences and noise.

The conventional non-adaptive beamforming is known to be optimal in the sense that it provides the highest possible output signal-to-noise ratio (SNR) and signal-to-interference plus noise ratio (SINR) in the background of white Gaussian noise [32]. The output SINR of the FDA radar can be evaluated by

$$ {\begin{aligned} \text{SINR} \backsimeq \frac{{\sigma^{2}_{s}} M^{2}}{\sum\limits_{i}{{\sigma^{2}_{i}}|\mathbf{b}^{H}(r_{0}, \theta_{0}) \mathbf{b}(r_{i}, \theta_{i})|^{2} |\mathbf{b}^{H}(r_{0}, \theta_{0}) \mathbf{b}(r_{i}, \theta_{i})|^{2}}+{\sigma^{2}_{n}} M}, \end{aligned}} $$

(29)

where \({\sigma ^{2}_{s}}\) is the variance of the desired target signal, \({\sigma ^{2}_{i}}\) is the variance of the *i*th interference, and \({\sigma ^{2}_{n}}\) is the noise variance. If the target is observed in the background of few weak interferences which are well separated from the target, the interference-to-noise power can be attributed to the noise term only. In this case, the SINR for the FDA radar simplifies to

$$ \text{SINR}\backsimeq\frac{{\sigma^{2}_{s}} M}{{\sigma^{2}_{n}}}, $$

(30)

which means that the FDA radar has an equivalent robustness again noise.

In contrast, if the target is observed in the background of strong interferences, then we can fairly consider the noise power to be negligible as compared to the interference power. In such a case, we have

$$ \begin{aligned} \text{SINR} \backsimeq\frac{{\sigma^{2}_{s}} M^{2}}{\sum\limits_{i}{{\sigma^{2}_{i}}|\mathbf{b}^{H}(r_{0}, \theta_{0}) \mathbf{b}(r_{i}, \theta_{i})|^{2} |\mathbf{b}^{H}(r_{0}, \theta_{0}) \mathbf{b}(r_{i}, \theta_{i})|^{2}}}. \end{aligned} $$

(31)

The FDA using Costas-sequence modulated frequency offsets will make the transmit-receive beampattern mainlobe approximate an ideal thumbtack response, as provided in the next section. In this case, the SINR (Eq. (31)) can be simplified as

$$ \text{SINR}\backsimeq\frac{{\sigma^{2}_{s}} M^{2}}{\overline{\sigma}^{2}_{i}}, $$

(32)

where \(\overline {\sigma }^{2}_{i}\) denotes the mean of \({\sigma ^{2}_{i}}\). Thus, it is expected that this FDA radar has better robustness against interferences than both conventional phased-array radar and standard FDA radar.